Supervisor: Professor Alistair Savage, Department of Mathematics & Statistics, University of Ottawa
Co-Supervisor: Dr. Yaolong Shen, Department of Mathematics & Statistics, University of Ottawa

Project Location: Fields Institute (Toronto, ON) - the project supervisors will be fully remote

Project Overview

This project will explore the algebraic and combinatorial properties of Laurent symmetric functions—generalizations of classical symmetric functions that allow both positive and negative powers of variables. These functions arise naturally in the representation theory of double affine Hecke algebras (DAHAs), Macdonald theory, and quantum toroidal algebras. The students will begin by studying the ring of Laurent symmetric functions and its bases, then investigate their role in the theory of Macdonald polynomials and affine Schur functions. A key goal is to understand how these functions encode representation-theoretic data and how they interact with operators from DAHAs. Depending on interest and progress, the project may include computational experiments or connections to Hilbert schemes and elliptic Hall algebras. This topic offers a rich blend of algebra, combinatorics, and geometry, and is well-suited for students interested in modern representation theory and algebraic combinatorics.

Project Keywords: Laurent symmetric functions, double affine Hecke algebras, Macdonald polynomials, representation theory

Supervisor: Professor Ricardo Baptista, Department of Statistical Sciences, University of Toronto
Co-Supervisor: Professor Farnaz Heidar-Zadeh, Department of Chemistry, Queen’s University

Project Location: Fields Institute (Toronto, ON)

Project Overview

Understanding the intrinsic dimension of molecules is essential for uncovering the structure of large chemical spaces, guiding representation learning, and improving generative molecular modeling. In this summer project, students will investigate the estimation of the intrinsic dimension of molecular data derived from both physical and learned latent representations. Following recent developments in dimension reduction procedures with theoretical guarantees, they will rigorously quantify the information lost from representing molecules in lower-dimensional spaces and show how the intrinsic dimension changes as a function of various quantities of interest for prediction. In addition to analytical quantifications of information loss, the students will implement methods to estimate latent representations for molecules and compare their quality to those derived based on the graphical structure, and physical and chemical constraints. Over the summer, the project will primarily focus on gold nano-clusters, which have recently emerged as promising agents in the diagnosis and treatment of cancer.

Project Keywords: Dimensionality Reduction, Computational Chemistry, Information Theory, Graph Theory, Latent Representations

Supervisor: Dr. Igor Gilitschenski, Department of Computer Science, University of Toronto

Project Location: Fields Institute (Toronto, ON)

Project Overview

The Toronto Intelligent Systems Lab (TISL), led by Professor Igor Gilitschenski, specializes in Computer Vision, Machine Learning, and Robotics. Successful projects will be supported in submission to some of the top-tier conferences in these fields (e.g. CVPR, ECCV, NeurIPS, ICML, ICLR, CoRL, RSS, ICRA).

We are generally looking for students with strong coding skills, as well as strong math fundamentals in linear algebra, calculus, optimization, probability, and statistics.We expect experience with at least one of the modern deep learning frameworks (PyTorch / JAX). This experience should ideally involve diving into existing codebases, debugging other people’s code, or (re-)implementing (rather than merely using) different neural network layer types. Ideally, you are also familiar with the industrial software development process (code reviews, unit testing, style guides, ...).

Project Keywords:

• Robotics: Data-driven Simulation, Vision-Language-Action models, Manipulation, Autonomous Driving.
• Computer Vision: Efficient & Editable Neural 3D Reconstruction, Generative Scene Representations, Controllable Video Generation, Neuromorphic Vision.
• Machine Learning: Geometric Deep Learning, Causal Representation Learning, Reinforcement Learning, Imitation Learning.

Supervisor: Dr. Logan Crew, Department of Combinatorics & Optimization, University of Waterloo
Co-Supervisor: Dr. Sophie Spirkl, Department of Combinatorics & Optimization, University of Waterloo

Project Location: Fields Institute (Toronto, ON) - the project supervisors will be fully remote

Project Overview

Directed graphs consist of a set of objects, and and arrows between some pairs of these objects. This very general construction models many real-world systems in chip design, network theory, particle physics, and more. A colouring splits the objects of a directed graph into sets where there is no cycle formed by arrows; such a partition of the objects has applications for avoiding conflicts in a variety of circumstances. Generally speaking, colouring problems help us to understand the structure of directed graphs, such as how local obstructions affect the entire system. This project aims to encode colouring data about directed graphs in the form of a polynomial, and study what information about the directed graph is determined by this data.

Project Keywords: graph theory, digraphs, tournaments, chromatic polynomial

Supervisor: Professor Mohamed Omar, Department of Mathematics & Statistics, York University

Project Location: Fields Institute (Toronto, ON)

Project Overview

This project explores a deep and exciting question in modern mathematics: how large can a set of numbers or geometric points be if it avoids particular patterns? Recent breakthroughs by Professor Omar used powerful new techniques—called slice-rank and partition-rank methods—that show such sets must often be much smaller than expected. These tools unify ideas from algebra, geometry, and combinatorics, and have solved problems going back to Paul Erdős, one of history's most influential mathematicians.

Undergraduate researchers will build on these discoveries to investigate new forbidden-pattern problems using accessible algebraic and combinatorial tools. Students will learn how structure emerges in large systems and contribute to advancing Euclidean Ramsey Theory, a field studying when order must appear in seemingly random settings. The project combines hands-on problem solving with exposure to cutting-edge methods, offering opportunities for original results and publication-level research. This is ideal for students curious about deep patterns hidden in mathematics.

Project Keywords: partition rank, slice rank, extremal combinatorics, Euclidean Ramsey Theory

Supervisor: Dr. Divya Sharma, Department of Mathematics & Statistics, York University
Co-Supervisor: Dr. Jude Kong, Department of Mathematics & Dalla Lana School of Public Health, University of Toronto

Project Location: Fields Institute (Toronto, ON)

Project Overview

This project introduces students to a new way of using mathematics to understand how diseases spread. Instead of relying only on equations like those in traditional SIR or SEIR epidemic models, students will learn how to build Physics-Informed Neural Networks (PINNs) — a type of artificial intelligence that learns from data while following the rules of mathematics. The students will start with simple models that describe how infections rise and fall, then train their PINN to match real or simulated outbreak data, even when that data is noisy or incomplete. Through this hands-on experience, they will see how math, data, and computing work together to make better predictions and interpret how factors like public behavior or vaccination policies affect disease spread. The expected outcome is a deeper understanding of how mathematical models and AI can jointly support public health forecasting and decision-making.

Project Keywords: physics-Informed Neural Networks (PINNs), epidemic modeling, SIR/SEIR models, differential equations, applied mathematics, scientific machine learning, data-driven modeling, epidemiology, public health forecasting, interpretability, adaptive behavior modeling, computational modeling, AI in health, parameter estimation, simulation and prediction

Supervisor: Dr. Taylor Brysiewicz, Department of Mathematics, Western University
Co-Supervisor: Dr. Graham Denham, Department of Mathematics, Western University

Project Location: Western University (London, ON)

Project Overview

• How many 3-point lines can be drawn through 9 points? How many 4-point circles?
• What is the dimension of the space of octahedra?
• Given a prescribed list of 3-point lines, like {123,124,234,145}, can one draw a realization with these collinearities?

At the center of these inquiries are mathematical objects called "configuration spaces". While it is not so hard to write down equations for these spaces, solving those equations quickly becomes a challenge. This project will launch a computational investigation of such objects using the techniques of a field of computational mathematics called 'numerical algebraic geometry'. We will develop a software package to study these objects and push our code to the limit to discover new results about these complicated spaces.

Project Keywords: geometry, combinatorics, algebraic geometry, computation, matroids, numerical algebraic geometry

Supervisor: Professor Camila de Souza, Department of Statistical & Actuarial Sciences, Western University
Co-Supervisors: Professor Andrea Soddu, Department of Physics & Astronomy, Western University; Professor Hanna Jankowski, Department of Mathematics & Statistics, York University

Project Location: Western University (London, ON)

Project Overview

Quantum Cognition Machine Learning (QCML) is a novel computational framework which brings together data structures from quantum mechanics with statistical and computational ML methods. The QCML framework is able to leverage the uncertainty inherent in quantum mechanics objects, thus improving data economy over classical statistical Machine Learning (ML). The method was developed at the company Qognitive, and as such, is not yet widely available. The proposed projects will (1) implement a QCML approach, (2) apply it to a tinnitus dataset, (3) time permitting, use simulations to understand the role of quantum geometry in QCML vs standard ML performance.

Project Keywords: statistical machine learning, quantum mechanics, neuroscience